For the ice cream, place lime zest and sugar in a food processor and pulse for 1 minute; add lime juice and process until blended. Pour mixture into a medium bowl; whisk in coconut milk, salt, and rum until smooth. Cover and chill for 4 hours in the refrigerator, then pour into frozen cylinder of ice cream maker; freeze according to manufacturer’s instructions. Spoon mixture into a freezer-safe container and place in the freezer until firm.

For the cookies, whisk together flours, salt, and baking powder in a medium bowl. Beat butter and brown sugar in a large mixer bowl at medium speed until creamy. Beat in eggs, vanilla, and honey until combined. Reduce speed to low; gradually add flour mixture and beat just until incorporated. Scrape dough onto a piece of plastic wrap and shape into a disc; wrap in plastic and chill in the refrigerator for at least two hours or up to overnight.

Preheat oven to 350°F and line two baking sheets with parchment paper. Divide the dough into 2 sections; working with one section at a time, roll dough out to ¼-inch thickness on a lightly floured board. Use a 2 ¼-inch fluted biscuit cutter to cut into circles. Repeat with remaining dough, re-rolling scraps as needed, until you have 48 circles.

Use a wooden skewer to poke a pattern of holes on half of the circles (these will be your tops). Combine cinnamon and sugar in a small bowl and sprinkle over cookies. Chill cookies on baking sheets in the freezer for 10 minutes.

Bake for 7 minutes, until cookies are slightly puffed and edges are firm. Cool for 3 minutes on baking sheets, then transfer to a wire rack to cool completely. Freeze cookies in an airtight container for at least one hour.

To assemble sandwiches, use a small cookie scoop to place a scoop of ice cream between two chilled cookies and press lightly. Place assembled ice cream sandwiches on a baking sheet and cover with plastic; return to the freezer for two hours. Wrap sandwiches individually in plastic wrap and store in the freezer for up to 7 days. Let stand at room temperature for 5 minutes before serving.